Towards a conjecture on the distance spectral radius of trees

Valisoa Razanajatovo Misanantenaina
Stellenbosch University

Stephan Wagner
Stellenbosch University

Kenneth Dadedzi
Stellenbosch University



Content: The distance matrix of a graph is the matrix whose entry in the $i$th row, $j$th column is the distance $d(v_i,v_j)$ between the $i$th vertex $v_i$ and the $j$th vertex $v_j$. A conjecture of Ili\'c and Stevanovi\'c states that among all trees with given order and maximum degree, the so-called Volkmann trees minimise the spectral radius of the distance matrix. In this talk, we present our recent progress towards this conjecture and its analogue for a ``reversed'' version of the distance matrix. Our ideas are based on similar results for the Wiener index and the spectral radius of the adjacency matrix.

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